• # question_answer 16)                  Find the product:                 (i) $(5-2x)\,(3+x)\,$      (ii) $(x+7y)\,(7x-y)$      (iii) $({{a}^{2}}+b)(a+{{b}^{2}})$                 (iv) $({{p}^{2}}-{{q}^{2}})\,(2p+q)$

(i) $(5-2x)\,(3+x)$                 $(5-2x)\,(3+x)$ $=(5)\times (3+x)-(2x)\times (3+x)$ $=(5)\times \left( 3 \right)\,+(5)\,\times (x)\,-(2x)$$\times (3)\,-(2x)\times (x)$ $=15+5x-6x-2{{x}^{2}}$ $=15-x-2{{x}^{2}}$                                        |Combining like terms (ii) $(x+7y)\,(7x-y)$ $=(x+7y)\,\times (7x-y)$ $=(x)\times (7x-y)+(7y)\,\times (7x-y)$ $=(x)\times (7x)-(x)\times (y)\,+(7y)$$\times (7x)-(7y)\times (y)$                 $=7{{x}^{2}}-xy+49yx-7{{y}^{2}}$ $=7{{x}^{2}}+48xy-7{{y}^{2}}$                 |Combining like terms (iii) $({{a}^{2}}+b)\,(a+{{b}^{2}})$ $({{a}^{2}}+b)\,(a+{{b}^{2}})$ $={{a}^{2}}\times \,(a+{{b}^{2}})\,+b\times \,(a+{{b}^{2}})$ $=({{a}^{2}})\times (a)+({{a}^{2}})\times ({{b}^{2}})+(b)$$\times (a)+(b)\times ({{b}^{2}})$ $={{a}^{3}}+{{a}^{2}}{{b}^{2}}+ba+{{b}^{3}}$ (iv) $({{p}^{2}}-{{q}^{2}})\,(2p+q)$ $({{p}^{2}}-{{q}^{2}})\,\times (2p+q)$ $={{p}^{2}}\times \,(2p+q)-{{q}^{2}}\times \,(2p+q)$ $=({{p}^{2}})\,\times (2p)+({{p}^{2}})\times (q)-({{q}^{2}})$$\times (2p)\,-({{q}^{2}})\times (q)$ $=2{{p}^{3}}+{{p}^{2}}q-2{{q}^{2}}p-{{q}^{3}}$